Transcript Waves and Osc. Chap2x

Damped Simple Harmonic
Oscillator
CHAPTER – 2
Mrs. Rama Arora
Assoc. Professor
Deptt. Of Physics
PGGCG-11
Chandigarh
 Damping is the mechanism that results in
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dissipation of the energy of an oscillator.
In case of damped simple harmonic vibrations
amplitude goes on decreasing with the passage of
time and ultimately the body comes to rest.
The damping factor for the mechanical oscillator is a
force.
Units of damping constant ‘r’ is kg/s or Ns/m.
In electrical oscillator damping factor is e.m.f.
The unit of R is ohms.
 In mechanical oscillator the damping force is due to
(i) viscous damping (ii) friction damping (iii)
structure damping
 In electrical oscillator the damping is due to
resistance in the circuit.
 Damping force is neither a constant nor depends
upon displacement or acceleration but depends upon
velocity only.
Differential equation of Simple Damped
Harmonic motion
 Restoring force is always proportional to the
displacement of the body.
 Damping force which is proportional to the velocity
of the body.
 Differential equation of damped SHM oscillator is:
 The solution is
 Term
is an exponentially decreasing term with
increasing time i.e. amplitude goes on decreasing
with time.
 For heavy damping
so damping makes the
system non oscillatory.
 For critical damping b2 = w2 . The displacement
decays to zero exponentially and the system
returns to the initial state in the minimum possible
time.
 For light damping b2 < w2; amplitude of the
damped oscillations reduces exponentially to zero.
The oscillations cease almost during the same time
in which oscillator returns to initial state.
 The frequency of damped oscillations is given by
Variation of x with time in Damped Oscillator
Comparison between undamped and damped
oscillations
Undamped oscillation
Damped oscillations
1. Motion is strictly
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periodic and simple
harmonic.
Amplitude is constant.
Frequency is
determined by inertia
and elastic properties.
No dissipation of
energy occurs.
Oscillations continue
indefinitely.
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5.
Motion is not strictly
periodic or simple
harmonic.
Amplitude decreases with
time.
Frequency is determined by
inertia, elastic properties
and the damping constant.
Energy is dissipated
continuously.
Oscillations cease after some
time.
Damped Electrical Oscillator
 In the damped electrical oscillator, the dissipation of
energy occurs in the resistance of the circuit which
may be distributed due to connecting wires and
inductor.
 Equation of damped electrical oscillator is
 The solution of equation of damped electrical
oscillator is
 Heavy damping b2 > w2, the charge on the capacitor
decays to zero in the minimum possible time.
 Critical damping b2 = w2; the discharge is nonoscillatory.
 The behaviors of the oscillator is said to be dead
beat.
 Light damping b2 < w2
 The discharge is oscillatory and frequency of the
damped oscillations is given by
 Logarithmic decrement of a damped oscillator is the
natural logarithm of the ratio of amplitude of
oscillation at any instant and that one time after it.
 For mechanical oscillator
 For electrical oscillator
 Relaxation time is defined as the time interval during
which the amplitude of damped oscillator decays to 1/e
times its initial value.
 For mechanical oscillator
Relaxation time is inversely proportional to the damping
constant.
 For electrical oscillator
Thus, the relaxation time is inversely proportional to the
resistance of the oscillatory circuit.
 Quality factor gives the rate of decay of energy of the
damped oscillator and is a number equal to 2
times the ratio of the instantaneous energy of the
oscillator to the energy lost during one time period
after that instant.
 i.e.
 Relation between logarithmic decrement,
relaxation time and Θ factor.